BTW why did you think $\mathrm{B} \cup \mathrm{C} =\mathrm{A}$ always?

The question just said $\mathrm{B} \subseteq \mathrm{A}$, $\mathrm{C} \subseteq \mathrm{A}$ and $\mathrm{B} \cap \mathrm{C}=\varnothing$.

The question just said $\mathrm{B} \subseteq \mathrm{A}$, $\mathrm{C} \subseteq \mathrm{A}$ and $\mathrm{B} \cap \mathrm{C}=\varnothing$.